Abstract

It is known that every Gδ subset E of the plane containing a dense set of lines, even if it has measure zero,
has the property that every real-valued Lipschitz function on
ℝ2 has a point of differentiability in E. Here
we show that the set of points of differentiability of
Lipschitz functions inside such sets may be surprisingly tiny:
we construct a Gδ set E⊂ℝ2 containing a dense set of lines for which there is a pair of
real-valued Lipschitz functions on ℝ2 having no
common point of differentiability in E, and there is a
real-valued Lipschitz function on ℝ2 whose set of
points of differentiability in E is uniformly purely unrectifiable.

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